Quantum spectral problems and isomonodromic deformations

Abstract

We develop a self-consistent approach to study the spectral properties of a class of quantum mechanical operators by using the knowledge about monodromies of 2 × 2 linear systems (Riemann-Hilbert correspondence). Our technique applies to a variety of problems, though in this paper we only analyse in detail two examples. First we review the case of the (modified) Mathieu operator, which corresponds to a certain linear system on the sphere and makes contact with the Painlevé III 3 equation. Then we extend the analysis to the 2particle elliptic Calogero-Moser operator, which corresponds to a linear system on the torus. By using the Kiev formula for the isomonodromic tau functions, we obtain the spectrum of such operators in terms of self-dual Nekrasov functions (ǫ 1 + ǫ 2 = 0). Through blowup relations, we also find Nekrasov-Shatashvili type of quantizations (ǫ 2 = 0).In the case of the torus with one regular singularity we obtain certain results which are interesting by themselves. Namely, we derive blowup equations (filling some gaps in the literature) and we relate them to the bilinear form of the isomonodromic deformation equations. In addition, we extract the ǫ 2 → 0 limit of the blowup relations from the regularized action functional and CFT arguments.

Exact solution of Kerr black hole perturbations via CFT$_2$ and instanton counting. Greybody factor, Quasinormal modes and Love numbers

“…where 1 is a parameter which stays finite in the large central charge limit and plays the rôle of the Planck constant. The advantage of this approach is that the explicit solution of the connection problem on the z-plane for equation (28) can be derived from the explicit computation of the full CFT 2 correlator (42) and from its expansions in different intermediate channels. A crucial ingredient to accomplish this program is a deep control on the analytic structure of regular and irregular Virasoro conformal blocks.…”

“…From the CFT 2 viewpoint, the gauge theoretical approach corresponds to the large Virasoro central charge limit recalled above. It would be interesting to explore the relation between the c = 1 and c = ∞ approaches (see [28] for recent interesting developments). Let us remark that in our view the CFT 2 framework is the suitable one to provide a physical explanation of the above described relations among black hole physics and supersymmetric gauge theories.…”

M2-branes and $\mathfrak{q}$-Painlevé equations

“…• The identification of the τ -function with the spectral determinant of the quantum operator implies that the analysis of the zeroes of the first solves the quantum spectrum of the latter [6,32]. Therefore, the results obtained in this paper provide a method to quantize the integrable spin chain systems associated to 5d gauge theories with N f ≤ 4.…”

“…Note that this Fredholm determinant is different from the spectral determinant of the auxiliary linear problem associated with the Painlevé equation. See for example[32].…”

QNMs of scalar fields on small Reissner-Nordström-AdS$\mathbf{_5}$ black holes

“…The Painlevé transcendents implement the Riemann-Hilbert map relating the accessory parameters of the differential equations to the monodromy properties of its solutions, and [15,20] showed that their general expansion is expressible in terms of c = 1 conformal blocks. We should mention at this point that an alternative proposal directly using four-dimensional supersymmetric gauge theories in the Nekrasov-Shatashvili phase, corresponding to semi-classical conformal blocks, was developed in [21,22]. Also worthy of note is the effort to relate quantities of physical interest to monodromy data in [23], where the authors considered greybody factors, QNMs and Love numbers for the fourdimensional Kerr background.…”